Related papers: Introduction to generalised Cesaro convergence I

Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequality for convex functions known as Sherman's inequality. We extend Sherman's result to the class of n-strongly convex functions using…

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number…

We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…

Sequential analysis encompasses simulation theories and methods where the sample size is determined dynamically based on accumulating data. Since the conceptual inception, numerous sequential stopping rules have been introduced, and many…

In this paper we present a method to deal with divergences in perturbation theory using the method of the Zeta regularization, first of all we use the Euler-Mc Laurin Sum formula to associate the divergent integral to a divergent sum in the…

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin…

A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…

We generalize the property that Riemann sums of a continuous function corresponding to equidistant subdivision of an interval converge to the integral of that function, and we give some applications of this generalization.

After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference…

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This…

We explore some convergence notions for set-convergence coming from modern summability methods. Specifically we will see the connections between Wijsman $f$-statistical convergence and Wijsman $f$-strong Ces\`aro convergence, when $f$ is a…

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel…

We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a…

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for…

Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper…

By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of…

Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…